Stability of poiseuille flow of a viscoplastic fluid with respect to perturbations of finite amplitude

The study made in [1] revealed that the Poiseuille flow of a viscoplastic fluid is stable with respect to infinitely small perturbations. At the same time, it is a known fact that at large Reynolds numbers a turbulent-flow regime of a viscoplastic fluid has been observed experimentally (see [2]). The divergence in the results from the linear theory of hydrodynamic stability of the experimental data indicates the need for investigating the stability of the Poiseuille flow of a viscoplastic fluid with respect to finite amplitude perturbations; this forms the main content of the present paper.     

Finite element solution of the Orr–Sommerfeld equation using high precision Hermite elements: plane Poiseuille flow

This paper presents a comprehensive review of the numerical techniques used during the past half century and their accuracy in hydrodynamic stability analysis of plane parallel flows. The paper also describes a finite element solution of the Orr–Sommerfeld equation using high precision Hermite elements. A stability analysis technique is performed by imposing an infinitesimal perturbation to the laminar base flow to determine the thresholds of neutral instabilities or the growth rate of the perturbation for any Reynolds and wave numbers. Validation of the present numerical technique is performed for plane Poiseuille flow. The numerical results, obtained with uniform and nonuniform meshes, show excellent agreement with the most accurate results available in the literature. Copyright © 2004 John Wiley & Sons, Ltd.     

涂布流动的非线性阈值问题

本文研究了沿斜面流动薄层液体的非线性稳定性,即涂布流动的非线性稳定性问题。我们将周恒对平面Poiseuille流提出的弱非线性理论应用于涂布流动。文中对自由表面的世界条件提出了一个合理的简化方法,对亚临界时不同Reynolds数及扰动频率,求出了有限扰动的阈值。     

ASYMPTOTIC ANALYSIS OF STABILITY PROBLEM OF PLANE POISEUILLE FLOW FOR HIGH REYNOLDS NUMBER

A study of the stabifity of plane Poiseuille flow at higher Reynolds number is made. Within a "triple-deck" structural framework, the qualitative behaviour of the eigenvalue of Orr-Sommerfeld equation is analytically obtained. The corresponding eigenfunction is formulated approximately.     

Asymptotic analysis of stability problem of plane poiseuille flow for high Reynolds number

A study of the stability of plane Poiseuille flow at higher Reynolds number is made. Within a “triple-deck” structural framework, the qualitative behavior of the eingenvalue of Orr-Sommerfeld equation is analytically obtained. The corresponding eigenfunction is formulated approximately.     

Direct numerical simulation of the amplification of a 2D temporal disturbance in plane Poiseuille flow

A direct numerical scheme is developed to study the temporal amplification of a 2D disturbance in plane Poiseuille flow. The transient non-linear Navier–Stokes equations are applied in a region of wavelength moving with the wave propagation speed. The complex amplitude involved in the perturbation functions is considered as the initial input of the non-linear stability equations. In this study a fully implicit finite difference scheme with five points in the flow direction and three points in the normal direction is developed so that numerical simulation of the amplification of a two-dimensional temporal disturbance in plane Poiseuille flow can be investigated. The growth and decay of the disturbance with time are presented and neutral stability curves which are in good agreement with existing solutions can be determined. The critical conditions as a function of the amplitude A₀ of the disturbance are presented. Fixing the wavelength, the Navier–Stokes equations are solved up to Re=10,000 a friction factor increasing with Reynolds number is observed. The 2D non-linear behaviour of the streamfunction, vorticity and velocity components at Re=10,000 are also exhibited. © 1998 John Wiley & Sons, Ltd.     

Linear stability of viscous incompressible flow in a circular viscoelastic tube

Flow stability in rigid tubes has been the subject of much research [1]. The overwhelming majority of authors of both theoretical and experimental studies now conclude that Poiseuille flow in a circular rigid tube is linearly stable. However, real tubes all possess elastic properties, the influence of which has not been investigated in such detail. For certain selected values of the parameters characterizing an elastic tube it has been shown that with respect to infinitesimal axisymmetric perturbations Poiseuille flow in the tube can be unstable [2]. In this case boundary conditions that did not take into account the fairly large velocity gradient of the undisturbed flow near the tube wall were used. The present paper reports the results of a numerical investigation of the linear stability of Poiseuille flow in a circular elastic tube with respect to three-dimensional perturbations in the form of traveling waves propagated along the system (azimuthal perturbation modes with numbers 0, 1, 2, 3, 4, and 5 are considered). It is shown that the elastic properties of the tube can have an important influence on the linear stability spectrum. In the case of axisymmetric perturbations it is possible to detect an instability which, at Reynolds numbers of more than 200, exists only for tubes whose modulus of elasticity is substantially less than that of materials in common use. The instability to perturbations of the second azimuthal mode is different in character, inasmuch as at Reynolds numbers greater than unity it occurs in stiffer tubes. Moreover, as the Reynolds number increases it can also occur in tubes of greater stiffness. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 126–134, November–December, 1986.     

Linear stability of two-dimensional steady flow in wavy-walled channels

Linear stability of fully developed two-dimensional periodic steady flows in sinusoidal wavy-walled channels is investigated numerically. Two types of channels are considered: the geometry of wavy walls is identical and the location of the crest of the lower and upper walls coincides (symmetric channel) or the crest of the lower wall corresponds to the furrow of the upper wall (sinuous channel). It is found that the critical Reynolds number is substantially lower than that for plane channel flow and that when the non-dimensionalized wall variation amplitude is smaller than a critical value (about 0.26 for symmetric channel, 0.28 for sinuous channel), critical modes are three-dimensional stationary and for larger , two-dimensional oscillatory instabilities set in. Critical Reynolds numbers of sinuous channel flows are smaller for three-dimensional disturbances and larger for two-dimensional disturbances than those of symmetric channel flows. The disturbance velocity distribution obtained by the linear stability analysis suggests that the three-dimensional stationary instability is mainly caused by local concavity of basic flows near the reattachment point, while the critical two-dimensional mode resembles closely the Tollmien–Schlichting wave for plane Poiseuille flow.     

The spectrum of small perturbations in plane couette-poiseuille flow

The spectrum of small perturbations of plane Couette-Poiseuille flow is studied. The perturbations are classified according to their behavior at large wave numbers. The changes in the spectrum are traced as the transition is made from Poiseuille to Couette flow at fixed Reynolds number. The behavior of the perturbations is considered as a function of the Reynolds number.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 2, pp. 63–67, March–April, 1971.The author wishes to thank M. A. Gol'dshtik for his attention to the paper, V. A. Sapozhnikov for useful discussions, and V. N. Shtern for his great assistance and help with the paper and for useful discussions.     

Finite-Amplitude Equilibrium Solutions for Plane Poiseuille–Couette Flow

Two-dimensional nonlinear equilibrium solutions for the plane Poiseuille–Couette flow are computed by directly solving the full Navier–Stokes equations as a nonlinear eigenvalue problem. The equations are solved using the two-point fourth-order compact scheme and the Newton–Raphson iteration technique. The linear eigenvalue computations show that the combined Poiseuille–Couette flow is stable at all Reynolds numbers when the Couette velocity component σ₂ exceeds 0.34552. Starting with the neutral solution for the plane Poiseuille flow, the nonlinear neutral surfaces for the combined Poiseuille–Couette flow were mapped out by gradually increasing the velocity component σ₂. It is found that, for small σ₂, the neutral surfaces stay in the same family as that for the plane Poiseuille flow, and the nonlinear critical Reynolds number gradually increases with increasing σ₂. When the Couette velocity component is increased further, the neutral curve deviates from that for the Poiseuille flow with an appearance of a new loop at low wave numbers and at very low energy. By gradually increasing the σ₂ values at a constant Reynolds number, the nonlinear critical Reynolds numbers were determined as a function of σ₂. The results show that the nonlinear neutral curve is similar in shape to a linear case. The critical Reynolds number increases slowly up to σ₂∼ 0.2 and remains constant until σ₂∼ 0.58. Beyond σ₂ > 0.59, the critical Reynolds number increases sharply. From the computed results it is concluded that two-dimensional nonlinear equilibrium solutions do not exist beyond a critical σ₂ value of about 0.59. Received: 26 November 1996 and accepted 12 May 1997     

Stability of plane-parallel flow of magnetic fluids under external magnetic flelds

In this work, we present a theoretical study on the stability of a twodimensional plane Poiseuille flow of magnetic fluids in the presence of externally applied magnetic flelds. The fluids are assumed to be incompressible, and their magnetization is coupled to the flow through a simple phenomenological equation. Dimensionless parameters are deflned, and the equations are perturbed around the base state. The eigenvalues of the linearized system are computed using a flnite difierence scheme and studied with respect to the dimensionless parameters of the problem. We examine the cases of both the horizontal and vertical magnetic flelds. The obtained results indicate that the flow is destabilized in the horizontally applied magnetic fleld, but stabilized in the vertically applied fleld. We characterize the stability of the flow by computing the stability diagrams in terms of the dimensionless parameters and determine the variation in the critical Reynolds number in terms of the magnetic parameters. Furthermore, we show that the superparamagnetic limit, in which the magnetization of the fluids decouples from hydrodynamics, recovers the same purely hydrodynamic critical Reynolds number, regardless of the applied fleld direction and of the values of the other dimensionless magnetic parameters.     

Investigation Of Turbulence at the Molecular Level

The influence of the molecular structure of gas flows on the characteristics of turbulent flows and the influence of the properties of molecules on turbulent processes have been studied. A review of the results of studies of turbulent processes is presented. Data on flows at the boundary of a supersonic jet and in a pipe with a divergent inlet section, and Hagen–Poiseuille flow are given. Experimental study of Hagen–Poiseuille flow has shown that the molecular properties of the medium have an effect on the critical Reynolds number. It is shown that in comparing the critical Reynolds numbers for flows of different gases at different pressures, the common determining parameter is the second virial coefficient.     

Linear stability of plane creeping Couette flow for Burgers fluid

It is well known that plane creeping Couette flow of UCM and Oldroy-B fluids are linearly stable. However, for Burges fluid, which includes UCM and Oldroyd-B fluids as special cases, unstable modes are detected in the present work. The wave speed, critical parameters and perturbation mode are studied for neutral waves. Energy analysis shows that the sustaining of perturbation energy in Poiseuille flow and Couette flow is completely different. At low Reynolds number limit, analytical solutions are obtained for simplified perturbation equations. The essential difference between Burgers fluid and Oldroyd-B fluid is revealed to be the fact that neutral mode exists only in the former.     

On Stability of Plane and Cylindrical Poiseuille Flows of Nanofluids

Stability of plane and cylindrical Poiseuille flows of nanofluids to comparatively small perturbations is studied. Ethylene glycol-based nanofluids with silicon dioxide particles are considered. The volume fraction of nanoparticles is varied from 0 to 10%, and the particle size is varied from 10 to 210 nm. Neutral stability curves are constructed, and the most unstable modes of disturbances are found. It is demonstrated that nanofluids are less stable than base fluids; the presence of particles leads to additional destabilization of the flow. The greater the volume fraction of nanoparticles and the smaller the particle size, the greater the degree of this additional destabilization. In this case, the critical Reynolds number significantly decreases, and the spectrum of unstable disturbances becomes different; in particular, even for the volume fraction of particles equal to 5%, the wave length of the most unstable disturbances of the nanofluid with particles approximately 20 nm in size decreases almost by a factor of 4.     

Étude analytique de perturbations de l'écoulement de Poiseuille dans un canalAnalytical study of Poiseuille flow perturbations in a channel

We present in this study an analytic solution, valid for intermediate Reynolds numbers, of the Poiseuille flow perturbation in a channel. We use a method based on the solution of a linearized form of perturbation equations. The analytic solutions allow us to determine the symmetric and antisymmetric eigenmodes. For any given entry velocity profile in the channel slightly perturbed from Poiseuille flow, the complete flow solution is obtained by using an appropriate orthonormalisation procedure for the bases of the two types of eigenfunctions. To cite this article: A. Hifdi, J. Khalid Naciri, C. R. Mecanique 332 (2004).     

Non linear transverse disturbances in plane poiseuille flow at low Reynolds numbers — Part I

Summary The behaviour and sign of the Reynolds stress for periodic perturbation of finite amplitude is studied in this paper for the plane Poiseuille flow. The case considered has Reynolds number 250 and wave number =1. The Reynolds stress that in the linear case was of opposite sign with respect to the viscosity, in the case considered becomes such for perturbation amplitudes which are still significant with respect to the dynamics of the mean flow. Some numerical results are given to characterize the phenomenon.

---

Sommario In questo lavoro si studia nel moto piano di Poiseuille il comportamento ed il segno dello stress di Reynolds per perturbazioni periodiche di ampiezza finita. E' trattato il caso del numero di Reynolds 250 e del numero d'onda =1. Si osserva che lo stress di Reynolds, che nel caso lineare era di segno opposto alla viscosità, diventa tale per ampiezza della perturbazione rilevante rispetto alla dinamica del moto medio. Sono dati alcuni risultati numerici che caratterizzano il fenomeno.

---

---

This work is part of a research program on hydrodynamics at the Istituto di Elaborazione dell'Informazione of the C.N.R., Pisa. The first Author has suggested and precisely formulated the problem, whereas the numerical work was carried out by the second Author.     

Three-dimensional instability of two counter-rotating vortices in a rectangular cavity driven by parallel wall motion

The linear stability of two counter-rotating vortices driven by the parallel motion of two facing walls in a rectangular cavity is investigated by a finite volume method. Critical Reynolds and wave numbers are calculated for aspect ratios ranging from 0.1 to 5. This range is sufficient to find the asymptotic behavior of the critical parameters when the aspect ratio tends to zero and infinity, respectively. The critical curve is smooth for all aspect ratios and, hence, the character of the instability changes continuously. When the moving walls are far apart the mechanism is centrifugal, as in the classical lid-driven cavity. For aspect ratios near unity a combined mechanism, also involving strain near the vortex cores, leads to the instability which tends to asymmetrically displace the vortex cores, very similar to the cooperative short-wave instability of a free counter-rotating vortex pair. In the limit when plane Poiseuille flow is approached in the bulk, the three-dimensional perturbations are strongly localized near both downstream ends of the moving walls.     

Super-stable parallel flows of multiple visco-plastic fluids

We consider the stability of a multi-layer plane Poiseuille flow of two Bingham fluids. It is shown that this two-fluid flow is frequently more stable than the equivalent flow of either fluid alone. This phenomenon of super-stability results only when the yield stress of the fluid next to the channel wall is larger than that of the fluid in the centre of the channel, which need not have a yield stress. Our result is in direct contrast to the stability of analogous flows of purely viscous generalised Newtonian fluids, for which short wavelength interfacial instabilities can be found at relatively low Reynolds numbers. The results imply the existence of parameter regimes where visco-plastic lubrication is possible, permitting transport of an inelastic generalised Newtonian fluid in the centre of a channel, lubricated at the walls by a visco-plastic fluid, travelling in a stable laminar flow at higher flow rates than would be possible for the single fluid alone.     

On the non-linear stability of parallel shear flows

The non-linear stability of parallel shear flows in incompressible fluids is studied by the Lyapunov method for stress-free boundary conditions. It is shown that plane Couette flows and plane Poiseuille flows are asymptotically stable for all Reynolds numbers.     

Convection mixte en fluide binaire avec effet Soret : étude analytique de la transition vers les rouleaux transversaux 2D

This Note deals with mixed convection in binary fluid with Soret effect in a rectangular duct heated from below. In particular, we study the transition towards transverse 2D rolls appearing at low Reynolds and Rayleigh numbers. The linear stability analysis of Poiseuille flow, with linearly stratified temperature and concentration fields, shows the influence of the separation ratio on the critical Rayleigh number at the transition towards the transversal 2D convective patterns. It highlights the presence, at low Reynolds numbers, of propagating transverse rolls in the downwards as well as in the upwards direction. Finally, we point out that, under these conditions, the propagating frequency of the rolls is the sum of two well defined frequencies: the first related to the Reynolds, the second to the separation ratio. To cite this article: E. Piquer et al., C. R. Mecanique 333 (2005).